1. Field of the Invention
This invention relates generally to communication transmitters, receivers, and systems. More particularly, the invention relates an improved signal mapping technique for use with communications signals that have inherent memory.
2. Description of the Related Art
It is known that the event error probability, PE, and the bit error probability, PB, of continuous phase frequency shift keying (CPFSK) signals, over an additive white Gaussian noise (AWGN) channel with two-sided power spectral density, N0/2, can be approximately expressed at high signal to noise ratio (SNR) as:
                                                        P              E                        =                                          k                E                            ⁢                              Q                (                                                                                                    d                        min                        2                                            ⁢                                              E                        b                                                                                    N                      0                                                                      )                                              ;                ⁢                                  ⁢                              P            B                    =                                    k              B                        ⁢                          Q              (                                                                                          d                      min                      2                                        ⁢                                          E                      b                                                                            N                    0                                                              )                                                          (        1        )            where, dmin is the normalized minimum distance, Eb is the bit energy, kE is the event error coefficient, kB is the bit error coefficient, and Q(∘) is the standard Q-function. For further background information related to the above, the reader is referred to the references below.    [1] SUNDBERG, C. E.: ‘Continuous phase modulation’, IEEE Communications Magazine, 1986, (24), pp. 25-38    [2] ANDERSON, J. B.: ‘Digital Transmission Engineering’ (IEEE Press, 1998)    [3] PROAKIS, J. G.: ‘Digital Communications’, Fifth Edition (Mc-Graw-Hill, 2008)    [4] FONSEKA, J. P., DOWLING, E. M. and TENG, C. C.: ‘Quadrature multiplexed CPM’, IEEE Trans., 2008, COM-56, pp. 1487-1497. (see also US Patent Application 2007/0092018A1).    [5] S. Pizzi and S. G. Wilson, “Convolutional coding combined with continuous phase modulation”, IEEE Trans. on Commun., COM-33, pp. 20-29, January 1985.
In [1-3], equation (1) was derived starting with an upper-bound obtained from a union bound that considers all merging events. The union bound is then approximated as a sum of contributions from merging events with the minimum Euclidean distance. This approximation is justified at higher SNR values because the contributions from merging events with larger distances are negligible at higher SNR. Hence, an event error probability, i.e. the probability of selecting an incorrect survival path during Viterbi decoding, PE, can be written as in (1) with kE representing the total number of merging events with minimum distance. The bit error probability, PB, can then be calculated from PE by considering the conditional probability, PB|E, of a bit error conditioned on an event error. Hence, it follows from equation (1) that kB=kEPB|E.
FIG. 1 shows a phase tree and a state transition logic diagram of minimum shift keying (MSK). MSK has a modulation index of h=0.5, and a squared normalized minimum distance of dmin2=2.0[1]. In FIG. 1a, there is one regular merging event that occurs at point A with minimum distance. Since the phase π=−π, the points B and C merge and represent a special merging event [1] that also has the minimum distance. Hence, MSK signals have two merging events (A and special merging event [B,C]) with minimum distance, and thus kE=2. When an event error associated with either of these merging events occurs, the bits of intervals T and 2T are both decoded incorrectly, and hence, the conditional probability is PB|E=1. Therefore, for regular MSK signals, kE=kB=2.
As can be seen from FIG. 1a and FIG. 1b, in ordinary MSK, only two of the four phase states can be occupied during any given interval. Specifically, phase states 0 and π can be occupied during each odd interval while phase states π/2 and 3π/2 can be occupied during each even interval. For this reason, MSK behaves like a two state scheme [1-3]. This two-state structure is also present in the improved mapping policy discussed below.
It would be desirable to have an improved mapping policy that could reduce the bit error coefficient, kB, of MSK, CPFSK and other modulation signals with inherent memory. It would be desirable if the improved mapping policy could significantly improve bit error rate performance particularly at lower to moderate SNR values.